Problem: The moon's illumination changes in a periodic way that can be modeled by a trigonometric function. On the night of a full moon, the moon provides about $0.25$ lux of illumination (lux is the SI unit of illuminance). During a new moon, the moon provides $0$ lux of illumination. The period of the lunar cycle is $29.53$ days long. The moon will be full on December $25$, $2015$. Note that December $25$ is $7$ days before January $1$. Find the formula of the trigonometric function that models the illumination $L$ of the moon $t$ days after January $1$, $2016$. Define the function using radians. $ L(t) = $
Solution: Let's start by writing a formula for the light provided by the moon $u$ days after a full moon. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. At $u = 0$, on the night of a full moon, the light provided by the moon is at its maximum of $0.25$ lux. Let's use a cosine function: cosine functions have a maximum at $u = 0$. The illumination provided by the moon has a period of $29.53$ days. Its midline is halfway between its maximum and minimum values, or $\dfrac{0.25 + 0}{2} = 0.125$ Its amplitude is half the difference between its maximum and minimum illumination, or $\dfrac{0.25 - 0}{2} = 0.125$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we can stretch it horizontally by a factor of ${\dfrac{29.53}{2\pi}}$, stretch it vertically by a factor of ${0.125}$, and move it up ${0.125}$ units: $ L(u) = {0.125}\cos\left({\dfrac{2\pi}{29.53}}u\right) + {0.125}$ Since January $1$ is $7$ days after December $25$, the day that is $t$ days after January $1$ is $t + 7$ days after December $25$. So $u = t + 7$ : $ L(t) = {0.125}\cos\left({\dfrac{2\pi}{29.53}}(t + 7)\right) + {0.125}$ The function $ L(t) = {0.125}\cos\left({\dfrac{2\pi}{29.53}}(t + 7)\right) + {0.125}$ has period $29.53$, amplitude $0.125$ and midline $y = 0.125$, and it has a maximum at $t = -7$, so it's a good model of the illumination provided by the moon.